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How can we explicitly calculate the norm of a differential form?

For example let $(X, \omega) $ be a complex manifold such that locally $$ \omega(z) =i\sum_{k,j} h_{k, j} (z) dz_k\wedge d\overline {z_j}\, , $$ and let $\alpha $ be a $(p, q)-$form. How we can calculate the norm of $\alpha $ with respect to the metric $\omega$?

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    $\begingroup$ This is a question of Hermitian linear algebra, no need for manifolds. In my experience, formally working out the linear algebra is the way to understand things. Then, you can port your computations to the tangent space of the manifold, which is a particular Hermitian vector space. $\endgroup$
    – Andy Sanders
    Commented Jul 20, 2020 at 19:43

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Let $h^{\overline{i}j}$ be the inverse components, so that $h^{\overline{i}j}h_{j\overline{k}}=\delta_{\overline{k}}^{\overline{i}}$. If $\alpha$ is a (2,1)-form then $|\alpha|^2=h^{\overline{i}j}h^{\overline{k}l}h^{\overline{p}q}\alpha_{jl\overline{p}}\overline{\alpha_{ik\overline{q}}}$. Follow the same pattern for a $(p,q)$-form.

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  • $\begingroup$ Do you have a reference on this! $\endgroup$
    – user161399
    Commented Jul 20, 2020 at 20:40
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    $\begingroup$ I can't understand any of the textbooks, so to learn hermitian and kahler geometry I had to work it out for myself. But Griffiths & Harris "Principles of Algebraic Geometry" and chapter 2 of Besse "Einstein Manifolds" are standard references $\endgroup$
    – Quarto Bendir
    Commented Jul 20, 2020 at 22:33

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